# On The Equivalence of Projections In Relative $\alpha$-Entropy and R\'{e}nyi Divergence

## TL;DR

This paper proves the equivalence of projection theorems and geometric structures related to relative α-entropy and Rényi divergence, unifying their theoretical frameworks in information geometry.

## Contribution

It establishes the equivalence of two projection theorems and their geometric interpretations for relative α-entropy and Rényi divergence, linking their theoretical foundations.

## Key findings

- Projection theorems are equivalent under a probability measure correspondence.
- Associated Pythagorean theorems are also equivalent.
- The Riemannian geometries derived from both divergences are shown to be equivalent.

## Abstract

The aim of this work is to establish that two recently published projection theorems, one dealing with a parametric generalization of relative entropy and another dealing with R\'{e}nyi divergence, are equivalent under a correspondence on the space of probability measures. Further, we demonstrate that the associated "Pythagorean" theorems are equivalent under this correspondence. Finally, we apply Eguchi's method of obtaining Riemannian metrics from general divergence functions to show that the geometry arising from the above divergences are equivalent under the aforementioned correspondence.